Rate of convergence in the central limit theorem for empirical processes

Let _n and be an empirical process and a generalized Brownian bridge, respectively, indexed by a class of real measurable functions. From the central limit theorem for empirical processes it follows that for allr0. In this paper, assuming the class to be countably determined, under certain conditions we obtain an estimate for some constantC. Vapnik-ervonenkis class and the indicators of lower left orthants provide examples of classes considered here.     

A FUNCTIONAL CENTRAL LIMIT THEOREMFOR NEGATIVELY ASSOCIATED SEQUENCE

Let(x1,j≥1)be a sequence of negatively associated random variables with ex1=o,ex^21＜∞.in this paper a functional central limit theorem for negatively associated random variables under some conditions withbout stationarity is proved which is the same as the results for positively associated random variables.     

Central limit theorem and almost sure central limit theorem for the product of some partial sums

In this paper, we give the central limit theorem and almost sure central limit theorem for products of some partial sums of independent identically distributed random variables.     

On a multivariate central limit theorem for stationary bilinear processes

In 1957, Parzen proved a central limit theorem for a class of scalar processes which he called multilinear processes. In the present paper only stationary bilinear processes are considered, but the theory is generalized to the multivariate case.     

On the rate of convergence in the central limit theorem and the type of the Banach space

Let E be a Banach space. Let ξ be a sequence of which goes to zero. Let X be a centered E-valued random variable, which is bounded. Let S_n be the sum of n independent copies of X. Assume that whenever X satisfies the CLT, we have.

$\text{Sup}\text{t ?R}\text{|P(}\text{6S}{}_{\mathrm{n}}\text{√n | ? t}\text{) ? μ ({x;6x6 ?})| ? ξ}{}_{\mathrm{n}}$

where μ is the (Gaussian) limit of the laws of S_n. Then E is type 2.     

The central limit theorem for stationary associated sequences

We study the problem of convergence in distribution of a suitably normalized sum of stationary associated random variables. We focus on the infinite variance case. New results are announced. This revised version was published online in June 2006 with corrections to the Cover Date.     

Refinement of the almost sure central limit theorem for associated processes

In this paper, for the partial sumsS _n of a stationary associated random process it is proved that the logarithmic averages converge almost surely. The asymptotic normality of the normalized difference between the logarithmic averages and their limiting value is established. Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp. 513–522, October, 2000.     

Some estimates in theL p metric in the central limit theorem for additive functions

The convergence of the value distributions of a normalized sequence of strongly additive arithmetic functions to the standard normal law in theL _p metric is considered. An asymptotic formula for the variance is obtained. In both cases, the remainders are expressed in terms of the third absolute moment of the additive function. Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 1, pp. 74–80, January–March, 1999. Translated by A. Mačiulis     

A functional central limit theorem for a generalized occupancy problem

A functional central limit theorem is proved for a class of finitely exchangeable random variables which are based on an occupancy scheme.     

Exact rates of convergence in some martingale central limit theorems

Renz [14], Ouchti [13], El Machkouri and Ouchti [3] and Mourrat [12] have established some tight bounds on the rate of convergence in the central limit theorem for martingales. In the present paper a modification of the methods, developed by Bolthausen [1] and Grama and Haeusler [7], is applied for obtaining exact rates of convergence in the central limit theorem for martingales with differences having conditional moments of order $2+\rho ,\rho >0$. Our results generalise and strengthen the bounds mentioned above.     

A note on applications of the martingale central limit theorem to random permutations

A unified martingale approach is presented for establishing the asymptotic normality of some sequences of random variables. It is applied to the numbers of inversions, rises, and peaks, respectively, as well as the oscillation and the sum of consecutive pair products of a random permutation. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 10, 323–332 (1997)     

A functional central limit theorem for a class of urn models

We construct an independent increments Gaussian process associated to a class of multicolor urn models. The construction uses random variables from the urn model which are different from the random variables for which central limit theorems are available in the two color case.     

The central limit theorem for stochastic processes II

The main result is that the necessary and sufficient conditions for the central limit theorem for centered, second-order processes given by Giné and Zinn⁽⁶⁾ can be obtained without any basic measurability condition. Furthermore we extend some of their results.     

An almost sure central limit theorem for self-normalized weighted sums

Let X, X1 , X2 , ··· be a sequence of nondegenerate i.i.d. random variables with zero means, which is in the domain of attraction of the normal law. Let {a ni , 1≤i≤n, n≥1} be an array of real numbers with some suitable conditions. In this paper, we show that a central limit theorem for self-normalized weighted sums holds. We also deduce a version of ASCLT for self-normalized weighted sums.     

A central limit theorem for random sums of random variables

Anscombe (1952) (also see Chung (1974)) has developed a central limit theoremof random sums of independent and identically distributed random variables. Applicability of this theorem in practice, however, is limited since the normalization requires random factors. In this paper we establish sufficient conditions under which the central limit theorem holds when such random factors are replaced by the underlying asymptotic mean and standard ddeviation. An application of this result in the context of shock models is also given.     

Some pairwise independent sequences for which the central limit theorem fails

Some simple examples are given of stationary, pairwise independent sequences of random variables for which the central limit theorem uterly fails     

The central limit theorem and chaos

Let X be a compact metric space studies some relationships between stochastic and f ： X→ X be a continuous map. This paper and topological properties of dynamical systems. It is shown that if f satisfies the central limit theorem, then f is topologically ergodic and / is sensitively dependent on initial conditions if and only if / is neither minimal nor equicontinuous.     

Rates of convergence in the operator-stable limit theorem

Suppose that the ^d -valued random vector is strictly operator-stable in the sense that , the characteristic function of , satisfies for everyt<0, for some invertible linear operatorB on ^d . Suppose also that for the i.i.d. random vectors {X _i } in ^d , . In the present paper, we study the rates of convergence of this operator-stable limit theorem in terms of several probability metrics. A new type of ideal metrics suitable for this rate-of-convergence problem is introduced.This research was partially supported by NSF, Grant DMS-9103452 and NATO, Grant CRG900798.